55
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Module 4:
Lecture 6 on Stress-strain relationship
and Shear strength of soils
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Contents
 Stress state, Mohr’s circle analysis and Pole, Principal
stress space, Stress paths in p-q space;
 Mohr-Coulomb failure criteria and its limitations,
correlation with p-q space;
 Stress-strain behavior; Isotropic compression and
pressure dependency, confined compression, large stress
compression, Definition of failure, Interlocking concept
and its interpretations, Drainage conditions;
 Triaxial behaviour, stress state and analysis of UC, UU, CU,
CD, and other special tests, Stress paths in triaxial and
octahedral plane; Elastic modulus from triaxial tests.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Principal stress relations at failure:
τ
σ = σ tan (45 + φ / 2)
+ 2c tan(45 + φ / 2)
σ = σ tan (45 − φ / 2)
− 2c tan(45 − φ / 2)
σ1
σ3
X
2
1
3
2
3
1
φ
σ3
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
90+φ
σ1 σ
Mohr-Coulomb Idealization of Geomaterials
σ 1′
σ1′-σ3′
σ 2′ = σ 3′
E′
σ 3′
σ 3′
ε1
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Mohr diagram and failure envelopes
Coulomb, in his investigations of retaining walls proposed
a relationship:
Where c is the inherent
shear strength, also
known as cohesion c
and φ is angle of
internal friction
The criterion contains two
material constants, c and φ, as
opposed to one material
constant for the Tresca
criterion
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Mohr Coulomb Yield/Failure Condition
Yielding (and failure) takes place in the soil mass when
mobilised (actual) shear stress at any plane (τm)
becomes equal to shear strength (τf) which is given by:
τm = c′+ σ′ntanφ′ = τf
where c′ and φ′ are strength parameters
f (σ′ )= τ - σ′n tanφ′– c′= 0
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Mohr-Coulomb Idealization of Geomaterials
Note that the value of
intermediate stress (σ2′)
does
not
influence
failure 
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Mohr-Coulomb Idealization of Geomaterials
By constructing a Mohr circle
tangent to the line (a stress
state associated with failure)
and
using
trigonometric
relations, the alternative form
of
τf = c + σf tan φ
in terms of principal stresses is
obtained:
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Relationship between Kf line and Mohr-Coulomb failure
envelope (in terms of principal stresses)
qf = a + pf tanΨ
Kf
Ψ
sinφ′ = tan Ψ
c′ = a/cosφ
a
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Mohr-Coulomb(MC) failure criterion
With no order implied by the principal stresses σ1, σ2,
σ3,the MC criterion can be written as:
Where,
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Mohr-Coulomb(MC) failure criterion
T0 is the theoretical MC uniaxial tensile strength that is not
observed in experiments; rather, a much lower strength T is
measured (σ1 = 0, σ2 = -T), with the failure plane being normal
to σ3. C0 is the theoretical MC uniaxial compressive strength.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Mohr-Coulomb(MC) failure criterion
 The shape of the failure surface in principal stress
space is dependent on the form of the failure
criterion: linear functions map as planes and
nonlinear functions as curvilinear surfaces.
 The following six equations below are represented by
six planes that intersect one another along six edges,
defining a hexagonal pyramid.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Pyramidal surface in principal stress space and
cross-section in the equi-pressure plane
This is the failure surface on the equipressure
(σ1+σ2+σ3= constant) or π-plane perpendicular to
thehydrostatic axis, where MC can be described
as an irregular hexagon with sides of equal length.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Mohr-Coulomb in Principal stress space:
Isotropy requires threefold
symmetry
because
an
interchange of σ1, σ2, σ3
should not influence the
failure surface for an
isotropic material. Note
that, the failure surface
need only be given in any
one of the 60° regions.
Mohr-Coulomb failure
surface
is
irregular
hexagon in principal
stress space.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Mohr-Coulomb in Principal stress space
Consider the transformation from principal stress space
(σ1, σ2, σ3) to the Mohr diagram (σ, τ).
 Although the radial distance from the hydrostatic
axis to the stress point is proportional to the
deviatoric stress, a point in principal stress space
does not directly indicate the value of shear stress on
a plane.
 However, each point on the failure surface in
principal stress space corresponds to a Mohr circle
tangent to the failure envelope.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Mohr-Coulomb in Principal stress space
 For the particular case where σ2 is the intermediate principal
stress in the order σ1 ≥ σ2 ≥ σ3, the failure surface is given by the
side ACD of the hexagonal pyramid. The principal stresses at
point D represent the stress state for a triaxial compression test
(σ1, σ2 = σ3)D, and point D is given by circle D in the Mohr
diagram.
 Similarly, for point C with principal stresses (σ3, σ1 = σ2)C
associated with a triaxial extension test, Mohr circle C depicts the
stress state. Points D and C can be viewed as the extremes of the
intermediate stress variation, and the normal and shear stresses
corresponding to failure are given by points Df and Cf.
 Points lying on the line CD (on pyramid failure surface) will be
represented by circles between C and D.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Mohr-Coulomb in Principal stress space:
σ1’
Mohr-Coulomb
It has corners that may
sometime create problems in
computations
However,
this
particular
difficulty is quite easily overcome
by introducing a local rounding
of the corners (Griffiths, 1990)
σ3’
σ2’
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Mohr-Coulomb model in p-q space
As described earlier,
F = (σ 1 '−σ 3 ' ) − (σ 1 '+σ 3 ' ) sin φ '−2c' cos φ ' = 0
σ 3 ' = σ 1 '−q
σ 1 ' = 3 p '−2σ 3 '
σ 1 ' = 3 p '−2σ 1 '+2q =
σ 3 ' = 3 p '−2σ 3 '−q =
(σ 1 '+σ 3 ' ) =
3 p '+2q
3
3 p '− q
3
6 p '+ q
3
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Mohr-Coulomb model in p-q space
Or,
6 p '+ q 
q = sin φ 
 + 2c cos φ
 3 
3q = 6 p ' sin φ + q sin φ + 6c cos φ
6 sin φ '
6c cos φ '
q=
p '+
(3 − sin φ ' )
(3 − sin φ ' )
So, Formulation for
Mohr-Coulomb model in
p-q space is,
F = q − ηp '−c * = 0
q = ηp '+ c *
where,
η=
6c cos φ '
*
6 sin φ '
and , c =
(3 − sin φ ' )
(3 − sin φ ' )
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Limitations of Mohr-Coulomb theory:
1. Linearization of the limit stress envelope
τ
Usual experimental range
in the laboratory
φ, c
σ
•
•
Possible overestimation of the safety factor in slope stability calculations,
Difficulties in calibration because of linearization
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Mohr circles for three dimensional state of stress
2. Effect of intermediate principal stress σ2 on condition
at failure.
 It is obvious that σ2 can have no influence on the
conditions at failure for the Mohr failure criterion, no
matter what magnitude it has.
 The intermediate principal stress σ2 probably does
have an influence in real soil, but the
Mohr‐Coulomb failure theory does not consider it.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
3. Mohr-Coulomb failure criterion is well proven for most
of the geomaterials, but data for clays is still
contradictory.
4. Soils on shearing exhibit variable volume change
characteristics
depending
on
pre-consolidation
pressure which cannot be accounted with MohrCoulomb theory.
5. In soft soils volumetric plastic strains on shearing are
compressive (negative dilation) whilst the MohrCoulomb model will predict continuous dilation.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Definition of failure :
Mohr-Coulomb failure criteria:
• Failure along a plane in a material occurs by critical
combination of normal and shear stress .
• τ = f(σ)
• τ = c + σtanφ
• Shear stress is function of material cohesion (c) and
angle of internal friction (φ)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Definition of failure :
Stress state cannot exist
τ
C
τ = f(σ)
τ = c + σtanφ
B
A
Shear failure occurs
Safe against failure
σ
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Flow Rule for Mohr Coulomb
For Mohr-Coulomb flow rule
is defined through the
‘dilatancy angle’ of the soil.
G(σ′)= τ - σ′n tanψ′ – const.= 0
where ψ′ is the dilatancy
angle and ψ′ ≤ φ′
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Interlocking concept and its interpretations :
Frictional soil behaviour is mainly influenced by two factors:
1. Frictional resistance between the soil particles.
2. To expand the soil against confining pressure (Dilatancy).
So, angular friction can be defined as:
φ= φu + β
3. where, φ is angle of sliding friction between mineral surfaces
and β is the effect of interlocking.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Interlocking concept and its interpretations :
φ = φu + β
•
φ varies with the nature of packing of the soil.
•
Denser the packing, higher is the value of φ .
•
If φu for a given soil is constant, β must change with the
denseness of the soil packing.
•
β increases with increasing denseness of the soil, because
more work to be done to overcome the effect of interlocking.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Interlocking concept and its interpretations :
Effect of angularity of soil particles:
•
Soil possessing angular soil particles will show
higher friction angle than that of rounded soil
particles.
•
Because angular soil particles will show
greater degree of interlocking and higher
value of β.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Interlocking concept and its interpretations : Dilation
φ= φu + Ψ
•
β is the function of dilatancy of the soil.
∆x
−∆y
dense
∆y
∆x
∆x
∆y
+∆y
loose
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Interlocking concept and its interpretations : Dilation
dense
∆x
∆y
•
•
loose
For loose sand the volume decreases with shearing
For dense sand the volume increases with shearing
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Interlocking concept and its interpretations : Dilation/ direct shear
response
Q/P
dense
loose
y
P
x
x
Q
y
dense
loose
x
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Interlocking concept and its interpretations : Dilation/ direct shear
response
Total work done,
dW = Pδy + Qδx
y
P
x
Pδy + Qδx = µPδx
δy/δx = µ – Q/P = - tanψ
tanψ = tanφm − tanφc , where, φc = tan-1µ
Alternatively,
φm = φc + ψ
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Q
How to understand dilatancy
i.e., why do we get volume changes when applying
shear stresses?
φ = ψ + φi
The apparent externally mobilized angle of friction on horizontal planes (φ)
is larger than the angle of friction resisting sliding on the inclined planes (φi).
strength = friction + dilatancy
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
How to understand dilatancy
Bolton, 1991
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
How to understand dilatancy
When soil is initially denser than the
critical state which it must achieve, then
as the particles slide past each other
owing to the imposed shear strain they
will, on average separate.
The particle movements will be spread
about mean angle of dilation Ψ
See the orientation 
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
How to understand dilatancy
 When soil is initially looser than the
final critical state, then particles will
tend to get closer together as the
soil is disturbed, and the average
angle of dilation will be negative,
indicating a contraction.
See the orientation 
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
How to understand dilatancy
 If the density of the soil does not have to
change in order to reach a critical state
then there is zero dilatancy as the soil
shears at constant volume.
 It is important to realise that a critical
state is only reached when the particles
have had full opportunity to juggle
around
and
come
into
new
configurations. If the confining pressure is
increased while the particles are being
moved around then they will tend to
finish up in a more compact state.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay